Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
(→sandbox) |
m (→sandbox) |
||
Line 75: | Line 75: | ||
</asy> | </asy> | ||
<cmath>\text{Prove the shaded areas are equal.}</cmath> | <cmath>\text{Prove the shaded areas are equal.}</cmath> | ||
− | |||
<asy> | <asy> | ||
− | + | for(int i = 0; i < 8; ++i){ | |
− | + | for(int j = 0; j < 8; ++j){ | |
− | + | filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle,black); | |
− | + | } | |
− | + | } | |
− | + | for(int i = 1; i < 7; ++i){ | |
− | + | filldraw((i,7-i)--(i+1,7-i)--(i+1,8-i)--(i,8-i)--cycle,white); | |
− | + | } | |
− | + | for(int i = 0; i < 5; ++i){ | |
− | + | filldraw((i,4-i)--(i+1,4-i)--(i+1,5-i)--(i,5-i)--cycle,white); | |
− | + | } | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | for(int i = 0; i < 5; ++i){ | |
+ | filldraw((8-i,4+i)--(7-i,4+i)--(7-i,3+i)--(8-i,3+i)--cycle,white); | ||
+ | } | ||
− | + | filldraw((1,0)--(2,0)--(2,1)--(1,1)--cycle,white); | |
− | + | filldraw((0,1)--(0,2)--(1,2)--(1,1)--cycle,white); | |
− | + | filldraw((7,8)--(6,8)--(6,7)--(7,7)--cycle,white); | |
− | + | filldraw((8,7)--(8,6)--(7,6)--(7,7)--cycle,white); | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
</asy> | </asy> | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− |
Revision as of 19:30, 25 April 2014
Bobthesmartypants's Sandbox
Solution 1
First, continue to hit
at
. Also continue
to hit
at
.
We have that . Because
, we have
.
Similarly, because , we have
.
Therefore, .
We also have that because
is a parallelogram, and
.
Therefore, . This means that
, so
.
Therefore, .
Solution 2
Note that is rational and
is not divisible by
nor
because
.
This means the decimal representation of is a repeating decimal.
Let us set as the block that repeats in the repeating decimal:
.
( written without the overline used to signify one number so won't confuse with notation for repeating decimal)
The fractional representation of this repeating decimal would be .
Taking the reciprocal of both sides you get .
Multiplying both sides by gives
.
Since we divide
on both sides of the equation to get
.
Because is not divisible by
(therefore
) since
and
is prime, it follows that
.
Picture 1
Picture 2